Abstract group 72.38: $Q_8\times C_3^2$

• Label: 72.38
• Order: \( 2^{3} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{2} \)
• Perm deg.: $14$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$Q_8\times C_3^2$
Order:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_4\times \GL(2,3)$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metacyclic (hence metabelian).

Group statistics

Order	1	2	3	4	6	12
Elements	1	1	8	6	8	48	72
Conjugacy classes	1	1	8	3	8	24	45
Divisions	1	1	4	3	4	12	25
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	4
Irr. complex chars.	36	9	0	45
Irr. rational chars.	4	17	4	25

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	6

Constructions

Presentation:	$\langle a, b \mid b^{12}=1, a^{6}=b^{6}, b^{a}=b^{7} \rangle$
Permutation group:	Degree $14$ $\langle(1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (9,11,10), (12,14,13), (1,2)(3,4)(5,6)(7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & -1 & -1 & -1 & 0 & 0 \\ 1 & -1 & 0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 2 & 0 & 4 \\ 5 & 3 & 4 \\ 6 & 6 & 6 \end{array}\right), \left(\begin{array}{rrr} 5 & 5 & 5 \\ 5 & 4 & 4 \\ 1 & 1 & 2 \end{array}\right), \left(\begin{array}{rrr} 3 & 5 & 2 \\ 0 & 2 & 0 \\ 4 & 6 & 3 \end{array}\right), \left(\begin{array}{rrr} 0 & 5 & 2 \\ 0 & 6 & 0 \\ 4 & 6 & 0 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Direct product:	$C_3$ ${}^2$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $C_6$ (12)	$C_2$ . $C_6^2$	$C_6$ . $(C_2\times C_6)$ (4)	$C_4$ . $(C_3\times C_6)$ (3)	all 6

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 36 subgroups, all normal (6 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3\times C_6$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_3^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_8\times C_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 72: $Q_8\times C_3^2$

Order 36: $C_3\times C_{12}$ x 3

Order 24: $C_3\times Q_8$ x 4

Order 18: $C_3\times C_6$

Order 12: $C_{12}$ x 12

Order 9: $C_3^2$

Order 8: $Q_8$

Order 6: $C_6$ x 4

Order 4: $C_4$ x 3

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 72: $Q_8\times C_3^2$

Order 36: $C_3\times C_{12}$

Order 24: $C_3\times Q_8$

Order 18: $C_3\times C_6$

Order 12: $C_{12}$

Order 9: $C_3^2$

Order 8: $Q_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 72: $Q_8\times C_3^2$ ($C_1$)

Order 36: $C_3\times C_{12}$ ($C_2$) x 3

Order 24: $C_3\times Q_8$ ($C_3$) x 4

Order 18: $C_3\times C_6$ ($C_2^2$)

Order 12: $C_{12}$ ($C_6$) x 12

Order 9: $C_3^2$ ($Q_8$)

Order 8: $Q_8$ ($C_3^2$)

Order 6: $C_6$ ($C_2\times C_6$) x 4

Order 4: $C_4$ ($C_3\times C_6$) x 3

Order 3: $C_3$ ($C_3\times Q_8$) x 4

Order 2: $C_2$ ($C_6^2$)

Order 1: $C_1$ ($Q_8\times C_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 72: $Q_8\times C_3^2$ ($C_1$)

Order 36: $C_3\times C_{12}$ ($C_2$)

Order 24: $C_3\times Q_8$ ($C_3$)

Order 18: $C_3\times C_6$ ($C_2^2$)

Order 12: $C_{12}$ ($C_6$)

Order 9: $C_3^2$ ($Q_8$)

Order 8: $Q_8$ ($C_3^2$)

Order 6: $C_6$ ($C_2\times C_6$)

Order 4: $C_4$ ($C_3\times C_6$)

Order 3: $C_3$ ($C_3\times Q_8$)

Order 2: $C_2$ ($C_6^2$)

Order 1: $C_1$ ($Q_8\times C_3^2$)

Series

Derived series

$Q_8\times C_3^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$Q_8\times C_3^2$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$Q_8\times C_3^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6$

$\lhd$

$Q_8\times C_3^2$

Supergroups

This group is a maximal subgroup of 48 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $25 \times 25$ rational character table.